For the third example, we consider real-world healthcare data of sleep patterns [1]}, [2]}.
Fig. REF  A illustrates time series regarding the abdominal respiratory
inductance plethysmography (RIP) and airflow of a person while sleeping.
We focus on the synchronization and interaction of RIP and airflow data.
We set \(y_1\)  as a variable describing RIP data and \(y_2\)  as that describing airflow data.
In addition, we set \(z_i(t)=y_i(t+1)\)  and \(X_i(t)=[y_i(t),z_i(t)]\)  for \(i=1,2\) .
We estimated the Koopman operator with the data and obtained the phase function and difference of the PCF \(\Gamma _d\)  in the same manner as the first and second examples.
The result is illustrated in Fig. REF  C.
There are two values, indicated by red points in Fig. REF  C, of \(\psi \)  which satisfies \(\Gamma _d(\psi )=0\)  and \(\frac{d\Gamma _d(\psi )}{d\psi }<0\) .
These values are stable fixed points of the phase difference \(\psi \) .
Thus, \(\psi \)  stays around one of the fixed points, and because of the effect of noise, \(\psi \)  sometimes slips to the other fixed point.
Indeed, Fig. REF  F shows the value of phase difference \(\psi =\theta _1-\theta _2\) .
The orange lines indicates the values of two stable fixed points.
We can see \(\psi \)  stays around two different values most of the time.
